Let us remind that a Puiseux series is a power series with rational exponents and with a minimum,
eventually negative, exponent. By the multivariate version of the Newton-Puiseux theorem [1]}, [2]}, [3]}
the solutions of (REF ) read
\(x_i(\epsilon ) = \sum _{k=k_{0i}}^{\infty } c_{ik} \epsilon ^{k/n},\) 
